Definitions

Levi

Levi

[lee-vahy, ley-vee; for 5 also lee-vee]
Levi, in the Bible. 1 Son of Jacob and Leah and eponymous ancestor of the Levites. His name appears infrequently—at his birth, when he and Simeon massacred the Shechemites out of revenge, when Jacob migrated to Egypt, and finally when he is named in the prophecy of his father. 2 See Matthew, Saint. 3, 4 Names in the Gospel genealogy.
Levi, Carlo, 1902-75, Italian writer and painter, noted as an anti-Fascist leader. After taking a medical degree, Levi devoted himself to painting, gaining international acclaim. His political activity in the 1920s resulted in his exile (1935-36) to the remote province of Lucania. His experiences there are described in Cristo si è fermato a Eboli (1945; tr. Christ Stopped at Eboli, 1947). While in France (1939-41) he wrote the essay Of Fear and Freedom (1946, tr. 1950). Levi's other works include The Watch (1948, tr. 1951) and The Linden Trees (tr. 1962), as well as studies of modern Italy, Sicily, and the USSR.
Levi, Edward Hirsch, 1911-2000, American lawyer, legal educator, and public official, b. Chicago, grad. Univ. of Chicago and Yale Univ. law school. Long associated with the Univ. of Chicago, he was a professor of law there (1945-75), founded (1958) its prestigious Journal of Law and Economics, was dean of the law school (1950-62), provost (1962-68), and president (1968-75). He served as President Gerald Ford's attorney general (1975-77) and did much to restore credibility to the position after the scandals of the Nixon years. He returned to the Univ. of Chicago as Glen A. Lloyd Distinguished Service Professor (1977-84). Among his writings are An Introduction to Legal Reasoning (1949), Four Talks on Legal Education (1952), and Point of View (1969).
Levi, Primo, 1919-87, Italian writer. A chemist of Jewish descent, Levi was sent to the concentration camp at Auschwitz during World War II. His first memoir, If This Is a Man (1947; also tr. as Survival in Auschwitz) is a restrained yet poignant testimony, devoid of rancor or protest, of the atrocities he witnessed. In his other autobiographical books, The Reawakening (1963; film, 1996) and the dark, posthumously published The Drowned and the Saved (1988), Levi relates the manner in which physical torture and annihilation were accompanied by a process of moral degradation. He stresses that survival was as much a spiritual quest to maintain human dignity as a physical struggle. The Periodic Table (1975), a collection of 21 meditations, each named for a chemical element, draws analogies between a young man's moral formation and the physical and chemical properties that circumscribe our humanity. Levi's novels include The Monkey's Wrench (1978) and If Not Now, When? (1986). He also wrote short stories, essays, and poetry. He died in a fall that was widely thought a suicide.

See his The Voice of Memory: Interviews 1961-1987 (2001), ed. by M. Belpoliti and R. Gordon; biographies by M. Anissimov (1996, tr. 1998), C. Angier (2002), and I. Thomson (2003).

Woodbury, Levi, 1789-1851, American cabinet officer and jurist, Associate Justice of the U.S. Supreme Court (1845-51), b. Hillsboro, co., N.H. Important as a politician and jurist in New Hampshire, he served as governor (1823-24) and as U.S. Senator (1825-31). President Andrew Jackson, whom he firmly supported, appointed (1831) him Secretary of the Navy. In 1834 when Henry Clay obtained the Senate's rejection of Roger B. Taney, who had been appointed in 1833, Woodbury was chosen U.S. Secretary of the Treasury and inherited the difficult task of transferring the government deposits from the Bank of the United States to state banks ("pet banks"). Successfully fulfilling his duties he continued as Secretary until the end of President Van Buren's term (1841). Again a Senator (1841-45), Woodbury was appointed to the Supreme Court by President Polk, and on the bench he generally concurred with the decisions of Chief Justice Taney. Many of his speeches and his writings (3 vol., 1852) have been published.

See D. B. Cole, Jacksonian Democracy in New Hampshire (1970).

Lincoln, Levi, 1749-1820, American public official, b. Hingham, Mass., grad. Harvard, 1772. A lawyer, he held various local offices during the American Revolution and later became a Jeffersonian political leader. He served (1801-4) as U.S. Attorney General. He was subsequently lieutenant governor and governor of Massachusetts.
Eshkol, Levi, 1895-1969, Israeli statesman, third prime minister of Israel, b. Ukraine; originally named Levi Shkolnik. In World War I he served in the Jewish Legion, which supported the British forces in Palestine. A leader in the Histradrut (General Federation of Jewish Labor) and the Mapai party, he served as Israel's minister of finance from 1952 to 1963, when he became prime minister. In 1965 he was challenged from within the Mapai party by David Ben-Gurion in a dispute over government policy. The party supported Eshkol, at which time Ben-Gurion and his followers, including Moshe Dayan, split from Mapai to form the Rafi party. Just prior to the Six-Day War (June, 1967), amid pressure for a more militant posture toward the Arab countries, Eshkol expanded the base of his coalition cabinet by including two new parties, Rafi and the right-wing Gahal; Rafi was represented by Moshe Dayan, who took over the ministry of defense. Eshkol died in office in 1969.

See his state papers, ed. by H. M. Christman (1969); biography by T. C. F. Prittie (1969).

Coffin, Levi, 1798-1877, American abolitionist, b. North Carolina. In 1826 he moved to the Quaker settlement of Newport (now Fountain City), Ind., where he kept a store until 1847. His home became a leading station of the Underground Railroad, of which he was styled "president."

See his Reminiscences (3d ed. 1898, repr. 1968).

(born July 31, 1919, Turin, Italy—died April 11, 1987, Turin) Italian writer and chemist. Two years after obtaining a degree in chemistry, Levi, who was Jewish, was captured by the Nazis and sent to Auschwitz as a slave labourer. His autobiographical works—If This Is a Man, or Survival in Auschwitz (1947), The Reawakening (1963), and The Drowned and the Saved (1986)—are restrained and moving accounts of and reflections on survival in the Nazi camps. His best-known work, The Periodic Table (1975), is a collection of 21 meditations, each named for a chemical element. The lingering effects of his wartime trauma may have led to his suicide.

Learn more about Levi, Primo with a free trial on Britannica.com.

(born July 31, 1919, Turin, Italy—died April 11, 1987, Turin) Italian writer and chemist. Two years after obtaining a degree in chemistry, Levi, who was Jewish, was captured by the Nazis and sent to Auschwitz as a slave labourer. His autobiographical works—If This Is a Man, or Survival in Auschwitz (1947), The Reawakening (1963), and The Drowned and the Saved (1986)—are restrained and moving accounts of and reflections on survival in the Nazi camps. His best-known work, The Periodic Table (1975), is a collection of 21 meditations, each named for a chemical element. The lingering effects of his wartime trauma may have led to his suicide.

Learn more about Levi, Primo with a free trial on Britannica.com.

(born Oct. 28, 1798, New Garden, N.C., U.S.—died Sept. 16, 1877, Cincinnati, Ohio) U.S. abolitionist. Despite little formal education, he became a teacher. As a devout Quaker, he opposed slavery. In 1826 he moved to Newport, Ind., where he made his home into a depot of the Underground Railroad and used much of his wealth as a merchant to help the escaping slaves. In 1847 he moved to Cincinnati, where he opened a store selling goods made only by free labour. He continued his work with the Underground Railroad until the outbreak of the American Civil War; he then worked to aid liberated slaves.

Learn more about Coffin, Levi with a free trial on Britannica.com.

(flourished 1st century AD, Palestine; Western feast day September 21, Eastern feast day November 16) One of the Twelve Apostles, traditional author of the first Gospel. According to the Gospels, he was a tax collector known as Levi when Jesus called him to be a disciple. Other information about him is scarce. The Gospel of Matthew is directed at a Jewish-Christian audience in a Jewish environment and may have been written originally in Hebrew, but it is now doubted that the apostle Matthew was its author. Tradition holds that Matthew conducted his ministry in Judaea, after which he served as a missionary to Ethiopia and Persia. Legend differs as to whether he died a martyr's death.

Learn more about Matthew, Saint with a free trial on Britannica.com.

(born Oct. 28, 1798, New Garden, N.C., U.S.—died Sept. 16, 1877, Cincinnati, Ohio) U.S. abolitionist. Despite little formal education, he became a teacher. As a devout Quaker, he opposed slavery. In 1826 he moved to Newport, Ind., where he made his home into a depot of the Underground Railroad and used much of his wealth as a merchant to help the escaping slaves. In 1847 he moved to Cincinnati, where he opened a store selling goods made only by free labour. He continued his work with the Underground Railroad until the outbreak of the American Civil War; he then worked to aid liberated slaves.

Learn more about Coffin, Levi with a free trial on Britannica.com.

The Levi-Civita symbol, also called the permutation symbol or antisymmetric symbol, is a mathematical symbol used in particular in tensor calculus. It is named after the Italian mathematician and physicist Tullio Levi-Civita.

Definition

In three dimensions, the Levi-Civita symbol is defined as follows:

varepsilon_{ijk} =
begin{cases} +1 & mbox{if } (i,j,k) mbox{ is } (1,2,3), (3,1,2) mbox{ or } (2,3,1), -1 & mbox{if } (i,j,k) mbox{ is } (3,2,1), (1,3,2) mbox{ or } (2,1,3), 0 & mbox{otherwise: }i=j mbox{ or } j=k mbox{ or } k=i, end{cases}

i.e. varepsilon_{ijk} is 1 if (i, j, k) is an even permutation of (1,2,3), −1 if it is an odd permutation, and 0 if any index is repeated.

An alternative notation for the three dimensional Levi-Civita symbol without case differentiation is

varepsilon_{ijk} = -[(i-j)^2%3][(i-k)^2%3][(j-k)^2%3][(j-(i%3)-frac{1}{2})^2-frac{5}{4}]

with % representing the modulo operator and i,j,k in lbrace1,2,3rbrace .

For example, in linear algebra, the determinant of a 3×3 matrix A can be written

det A = sum_{i,j,k=1}^3 varepsilon_{ijk} a_{1i} a_{2j} a_{3k}

(and similarly for a square matrix of general size, see below)

and the cross product of two vectors can be written as a determinant:

mathbf{a times b} = begin{vmatrix} mathbf{e_1} & mathbf{e_2} & mathbf{e_3}
a_1 & a_2 & a_3 
b_1 & b_2 & b_3 
end{vmatrix}

sum_{i,j,k

1}^3 varepsilon_{ijk} mathbf{e_i} a_j b_k or more simply:
mathbf{a times b} = mathbf{c}, c_i = sum_{j,k=1}^3 varepsilon_{ijk} a_j b_k.

According to the Einstein notation, the summation symbol may be omitted.

The tensor whose components in an orthonormal basis are given by the Levi-Civita symbol (a tensor of covariant rank n) is sometimes called the permutation tensor. It is actually a pseudotensor because under an orthogonal transformation of jacobian determinant −1 (i.e., a rotation composed with a reflection), it acquires a minus sign. Because the Levi-Civita symbol is a pseudotensor, the result of taking a cross product is a pseudovector, not a vector.

Note that under a general coordinate change, the components of the permutation tensor get multiplied by the jacobian of the transformation matrix. This implies that in coordinate frames different from the one in which the tensor was defined, its components can differ from those of the Levi-Civita symbol by an overall factor. If the frame is orthonormal, the factor will be ±1 depending on whether the orientation of the frame is the same or not.

Relation to Kronecker delta

The Levi-Civita symbol is related to the Kronecker delta. In three dimensions, the relationship is given by the following equations:

varepsilon_{ijk}varepsilon_{lmn} = det begin{bmatrix} delta_{il} & delta_{im}& delta_{in} delta_{jl} & delta_{jm}& delta_{jn} delta_{kl} & delta_{km}& delta_{kn} end{bmatrix}
= delta_{il}left(delta_{jm}delta_{kn} - delta_{jn}delta_{km}right) - delta_{im}left(delta_{jl}delta_{kn} - delta_{jn}delta_{kl} right) + delta_{in} left(delta_{jl}delta_{km} - delta_{jm}delta_{kl} right) ,
sum_{i=1}^3 varepsilon_{ijk}varepsilon_{imn} = delta_{jm}delta_{kn} - delta_{jn}delta_{km} ("contracted epsilon identity") (In Einstein notation, the duplication of the i index implies the sum on i. The previous is then noted: varepsilon_{ijk}varepsilon_{imn} = delta_{jm}delta_{kn} - delta_{jn}delta_{km})

sum_{i,j=1}^3 varepsilon_{ijk}varepsilon_{ijn} = 2delta_{kn}

Generalization to n dimensions

The Levi-Civita symbol can be generalized to higher dimensions:
varepsilon_{ijkldots} =
left{ begin{matrix} +1 & mbox{if }(i,j,k,l,dots) mbox{ is an even permutation of } (1,2,3,4,dots) -1 & mbox{if }(i,j,k,l,dots) mbox{ is an odd permutation of } (1,2,3,4,dots) 0 & mbox{if any two labels are the same} end{matrix} right.

Thus, it is the sign of the permutation in the case of a permutation, and zero otherwise.

Furthermore, for any n the property

sum_{i,j,k,dots=1}^n varepsilon_{ijkdots}varepsilon_{ijkdots} = n! follows from the facts that (a) every permutation is either even or odd, (b) (+1)2 = (-1)2 = 1, and (c) the permutations of any n-element set number exactly n!.

In index-free tensor notation, the Levi-Civita symbol is replaced by the concept of the Hodge dual.
In general n dimensions one can write the product of two Levi-Civita symbols as:

varepsilon_{ijkdots}varepsilon_{mnldots} = det begin{bmatrix}
delta_{im} & delta_{in} & delta_{il} & dots delta_{jm} & delta_{jn} & delta_{jl} & dots delta_{km} & delta_{kn} & delta_{kl} & dots vdots & vdots & vdots end{bmatrix} .

Now we can contract m indices. This will add a factor of m! to the determinant and we need to omit the relevant Kronecker delta.

Properties

(in these examples, superscripts should be considered equivalent with subscripts)

1. When n=2, we have for all i,j,m,n in {1,2},

varepsilon_{ij} varepsilon^{mn} = delta_i^m delta_j^n - delta_i^n delta_j^m, (1)
varepsilon_{ij} varepsilon^{in} = delta_j^n, (2)
varepsilon_{ij} varepsilon^{ij}=2. (3)

2. When n=3, we have for all i,j,k,m,n in {1,2,3},

varepsilon_{jmn} varepsilon^{imn}=2delta^i_j, (4)
varepsilon_{ijk} varepsilon^{ijk}=6, (5)
varepsilon_{ijk} varepsilon^{imn}=delta^{m}_jdelta^{n}_k - delta^{n}_jdelta^{m}_k. (6)

Proofs

For equation 1, both sides are antisymmetric with respect of ij and mn. We therefore only need to consider the case ineq j and mneq n. By substitution, we see that the equation holds for varepsilon_{12} varepsilon^{12}, i.e., for i=m=1 and j=n=2. (Both sides are then one). Since the equation is antisymmetric in ij and mn, any set of values for these can be reduced to the above case (which holds). The equation thus holds for all values of ij and mn. Using equation 1, we have for equation 2 varepsilon_{ij}varepsilon^{in} = delta_i^i delta_j^n - delta^n_i delta^i_j

= 2 delta_j^n - delta^n_j

= delta_j^n.

Here we used the Einstein summation convention with i going from 1 to 2. Equation 3 follows similarly from equation 2. To establish equation 4, let us first observe that both sides vanish when ineq j. Indeed, if ineq j, then one can not choose m and n such that both permutation symbols on the left are nonzero. Then, with i=j fixed, there are only two ways to choose m and n from the remaining two indices. For any such indices, we have varepsilon_{jmn} varepsilon^{imn} = (varepsilon^{imn})^2 = 1 (no summation), and the result follows. Property (5) follows since 3!=6 and for any distinct indices i,j,k in {1,2,3}, we have varepsilon_{ijk} varepsilon^{ijk}=1 (no summation).

Examples

1. The determinant of an ntimes n matrix A=(a_{ij}) can be written as

det A = varepsilon_{i_1cdots i_n} a_{1i_1} cdots a_{ni_n},

where each i_l should be summed over 1,ldots, n.

Equivalently, it may be written as

det A = frac{1}{n!} varepsilon_{i_1cdots i_n} varepsilon_{j_1cdots j_n} a_{i_1 j_1} cdots a_{i_n j_n},

where now each i_l and each j_l should be summed over 1,ldots, n.

2. If A=(A^1, A^2, A^3) and B=(B^1, B^2, B^3) are vectors in R^3 (represented in some right hand oriented orthonormal basis), then the ith component of their cross product equals

(Atimes B)^i = varepsilon^{ijk} A^j B^k.

For instance, the first component of Atimes B is A^2 B^3-A^3 B^2. From the above expression for the cross product, it is clear that Atimes B = -Btimes A. Further, if C=(C^1, C^2, C^3) is a vector like A and B, then the triple scalar product equals

Acdot(Btimes C) = varepsilon^{ijk} A^i B^j C^k.

From this expression, it can be seen that the triple scalar product is antisymmetric when exchanging any adjacent arguments. For example, Acdot(Btimes C)= -Bcdot(Atimes C).

3. Suppose F=(F^1, F^2, F^3) is a vector field defined on some open set of R^3 with Cartesian coordinates x=(x^1, x^2, x^3). Then the ith component of the curl of F equals

(nabla times F)^i(x) = varepsilon^{ijk}frac{partial}{partial x^j} F^k(x).

Notation

A shorthand notation for anti-symmetrization is denoted by a pair of square brackets. For example, for an n x n matrix, M,

M_{[ab]} = frac{1}{2}varepsilon_{abc} varepsilon^{cde} M_{de } = , frac{1}{2}(M_{ab} - M_{ba})

and for a rank 3 tensor T,

T_{[abc]} = , frac{1}{3!}(T_{abc}-T_{acb}+T_{bca}-T_{bac}+T_{cab}-T_{cba}).

Tensor density

The Levi-Civita symbol may be considered to be a tensor density in two different ways. It may be regarded as a contravariant tensor density of weight +1 or as a covariant tensor density of weight -1. In four dimensions,
epsilon^{alpha beta gamma delta} = epsilon_{alpha beta gamma delta} ,.

See also

References

  • Charles W. Misner, Kip S. Thorne, John Archibald Wheeler, Gravitation, (1970) W.H. Freeman, New York; ISBN 0-7167-0344-0. (See section 3.5 for a review of tensors in general relativity).


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